A decades-long legislative battle over New Jersey adoption records appears to be coming to an end.

On Monday, Republican Gov. Chris Christie conditionally vetoed a bill to permit adult adoptees to get their original birth certificates, but his instructions on how to fix the bill are likely to be accepted by lawmakers, advocates said.

This means a law should be passed this spring that will permit adults adopted in New Jersey to request birth documents with their birth mothers’ names.

“It’s a huge, huge victory, not only for adoptees, but for birth parents,” said Judy Foster, board member of the American Adoption Congress, an adoption reform organization.

Opponents, however, were disappointed.

“If this proposal is quote ‘adopted’ — and that is a play on words — we would advise women seeking privacy in adoption to give birth in a state that values privacy,” said Marie Tasy, executive director of the New Jersey Right to Life.

“It doesn’t seem fair to us that a woman can choose abortion and her identity remains private, but a woman who chooses adoption would not be able to protect her identity,” she said.

Under current state law, persons adopted in New Jersey need a court order to get their original birth documents. The state has about 150,000 sealed adoption records, said Ms. Foster, who surrendered a child for adoption in 1961 but has been reconnected to the adult child for 16 years.

According to Mr. Christie’s conditional veto of S873, original birth certificates will be issued to adult adoptees beginning Jan. 1, 2017.

During the interim period, birth parents who wish to remain unidentified can request that their names be redacted from the birth certificates.

Birth parents will also be able to specify whether they want no contact from the children they placed for adoption, or if they welcome contact directly or through an intermediary.

New Jersey state lawmakers are expected to approve Mr. Christie’s revisions within a few weeks, said Pam Hasegawa of the New Jersey Coalition for Adoption Reform and Education. Having a bipartisan compromise after more than 30 years of fighting over bills “is such a huge relief,” said Ms. Hasegawa, herself an adoptee.

Ms. Tasy was not so convinced: “We can’t be sure” of what the lawmakers will do, she said.

The open-records battle stems from the routine practice of issuing adopted children an amended birth certificate listing their adoptive parents as their legal parents. The original birth certificate is typically sealed, and cannot be divulged without a court order.

In the 1970s, adult adoptees began protesting the sealing of these records, saying they had the right to know their own heritage and identity. But opponents, including adoption agencies, pro-life and other groups, fought open-records reforms, saying these birth records needed to stay sealed to protect the privacy of birth mothers, most of whom were unwed when they gave birth.

Patrick Brannigan of the New Jersey Catholic Conference was unavailable to talk about Mr. Christie’s proposed law Monday, but in past testimony, he has quoted a court ruling that said adoptive parents “have an interest in having the birth records placed under seal” to permit them to raise the child as their own “without fear of interference from the natural parents.”

Open-records advocates contend that such fears are often overblown.

When such states as Delaware, Oregon and Tennessee opened their adoption records to adult adoptees, hundreds of birth parents consented to contact and only a “minuscule” number blocked release of the information or refused contact with the child, said Ms. Foster.

Other states likely to consider open-records bills this year include Pennsylvania, New York and Connecticut, she said.